A007121 -id:A007121 - cp's OEIS frontend

A355607 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 + x)^(x^k).

1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, -3, 0, 1, 0, 0, 6, 20, 0, 1, 0, 0, 0, -12, -90, 0, 1, 0, 0, 0, 24, 40, 594, 0, 1, 0, 0, 0, 0, -60, 180, -4200, 0, 1, 0, 0, 0, 0, 120, 240, -1512, 34544, 0, 1, 0, 0, 0, 0, 0, -360, -1260, 11760, -316008, 0, 1, 0, 0, 0, 0, 0, 720, 1680, 28224, -38880, 3207240, 0

Offset: 0

Seiichi Manyama, Jul 09 2022

			Square array begins:
  1,   1,   1,   1,    1,   1, 1, ...
  1,   0,   0,   0,    0,   0, 0, ...
  0,   2,   0,   0,    0,   0, 0, ...
  0,  -3,   6,   0,    0,   0, 0, ...
  0,  20, -12,  24,    0,   0, 0, ...
  0, -90,  40, -60,  120,   0, 0, ...
  0, 594, 180, 240, -360, 720, 0, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=1..4 give A007113, A007121, (-1)^n * A353229(n), A354625.

Cf. A292892, A355609, A355619.

T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 1)/(n-k*j)!);

T(0,k) = 1 and T(n,k) = -(n-1)! * Sum_{j=k+1..n} (-1)^(j-k) * j/(j-k) * T(n-j,k)/(n-j)! for n > 0.

T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling1(n-k*j,j)/(n-k*j)!.

A354625 Expansion of e.g.f. (1 + x)^(x^4).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, -360, 1680, -10080, 72576, 1209600, -14256000, 159667200, -1902700800, 24458353920, -120860812800, -193037644800, 23690780467200, -646842994237440, 14916006359654400, -230812655044608000, 3182953434006528000, -37667817509059584000

Offset: 0

Seiichi Manyama, Jul 09 2022

sign

Column k=4 of A355607.

Cf. A007121, A354624.

my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^x^4))

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^4*log(1+x))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-(i-1)!*sum(j=5, i,(-1)^j*j/(j-4)*v[i-j+1]/(i-j)!)); v;

a(n) = n!*sum(k=0, n\5, stirling(n-4*k, k, 1)/(n-4*k)!);

a(0) = 1; a(n) = -(n-1)! * Sum_{k=5..n} (-1)^k * k/(k-4) * a(n-k)/(n-k)!.

a(n) = n! * Sum_{k=0..floor(n/5)} Stirling1(n-4*k,k)/(n-4*k)!.

A355605 Expansion of e.g.f. (1 + x)^(x^2/2).

Original entry on oeis.org

1, 0, 0, 3, -6, 20, 0, -126, 1260, -4320, 5040, 180180, -2601720, 31309200, -372756384, 4877195400, -70178799600, 1099333347840, -18429818232960, 327676010785200, -6146676161388000, 121301442091851840, -2512746856371628800, 54527094987619716000

Offset: 0

Seiichi Manyama, Jul 09 2022

sign

Cf. A007121, A355603.

my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^(x^2/2)))

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2/2*log(1+x))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-(i-1)!/2*sum(j=3, i, (-1)^j*j/(j-2)*v[i-j+1]/(i-j)!)); v;

a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 1)/(2^k*(n-2*k)!));

a(0) = 1; a(n) = -(n-1)!/2 * Sum_{k=3..n} (-1)^k * k/(k-2) * a(n-k)/(n-k)!.

a(n) = n! * Sum_{k=0..floor(n/3)} Stirling1(n-2*k,k)/(2^k * (n-2*k)!).

cp's OEIS Frontend

A355607 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 + x)^(x^k).

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A354625 Expansion of e.g.f. (1 + x)^(x^4).

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A355605 Expansion of e.g.f. (1 + x)^(x^2/2).

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